Residence time distribution of the TRAM recycle reactor system

Shorter Commumcatlons The deactlvanon these curves The deactlvatlon

rate (-daldf) function,

IS

evaluated

by dertvatlon

of

Y(T, pA), IS calculated from eqn (5)

&partamento Unrversniad Apdo 644 Bdbao Espaia

1449 de Qutmrca de Bdbao

TLcnrca

J BILBAO

NOTATION

Its values are listed III Table 4 Equation (5) can be rearranged rnto

1+ Kan!2

P.?+’ p==* Q(T,PA)

KA + KX - K,rl2 Pn

(11)

k&X

k&x

where by use was made of pn = (w-p&/2 Plotting the data according to eqn (ll), a good linear relauonship IS obtamed for n = 1 (Fig 5) % and KS values are m agreement with the ones obtamed from the dlfferentlal reactor

05

I

1

I

I

I

06

07

09

09

10

&ldlll) Fa

It can be concluded C-Q fits equation --=da dt

5

PA*/W T, PA) vs PA

that the deactlvatlon

of catalyst

A-10-0 S-

kdKXp,: ~+(KA+KBL)PA+K.~R’

(12)

wn.h the lunetic constants even m Table 2 Equation (12) together vvlth eqns (2) and (3) descnbe at any reaction tune Deportamento de Quimrca Unlversuiad de Zaragoza

Technrca

benzyl alcohol, bcnzaldehyde and hydrogen, respecuvely a catalyst activity, eqn (3) concentration of benzyl alcohol chemlsorbed on C.W the catalyst to yield P, the polsonmg product FAD feed rate, mol A/hr KA, KR, KS eqmhbrmm adsorption constants for benzyl aIcohol, benzaldehyde and hydrogen m the main reactron eqmbbnum adsorptlon constant for benzyl alcohol KX to yield P In the deactlvanon reachon k dehydrogenatlon rate constant deactivation rate constant t P polybenzylene, poisoning matenal PA. PR, Ps benzyl alcohol, benzaldehyde and hydrogen partmI pressure m the thnd. atm PA0 benzyl alcohol partial pressure at the reactor entrance, atm reaction rate, mollg cat h (- r,4) dehydrogenauon rdem, at zero tune and at the same composmon m (-r&0 the reactant nnxture reactor, feeding (- r&O rdem, at zero time, m daerenual pure A T temperature, “K time, mm catalyst we&t, g r: conversion, referred to benzyl alcohol, at the reacX4, tor exit Idem, at zero tune (X&O deactwahon function for a chemical reaction con‘WT, PA) trolled by the surface reaction on one active ate, integral reactor ‘WT, PA) rdem, ddferentud reactor 7r total pressure A.R,S

the system

I CORELLA J M ASUA

ZaMgoza

Espaia

Corella J , Jodra L Cl and Romero A An Q&m 1976 72 823 Asua J M , Ph D Thesis Universuiad de Zaragoza 1978 r31 Corella J , Bdbao J and Sancho M H J Cat To be publrshed 141 Romero A, Ph D Thesis Universalad de Madnd 1973 151 Jodra L G and Corella J , An. Qwm 1974 70 9 Jodra L G , Corella J and Romero A , Spamsh Patent 55 040 R Bdbao I, Ph D Thesis Umversnlad de Bdbao 1977 1%8 7(3) 509 PI Chu Ch , I E C Fundament 191 Froment G F , PIWC of 6th Int Gong on Catalysts, London I976

[6

Residence time distributnonof the TRAM recycle reactor systemt (Recerved

for publrcatron

The tubular reactor vessel wtth axial murmg (TRAM) in a recycle loop has been used as a reactor vessel model for several analytIcal strokes m the hterature These studies were usually concertWork performed IL60616,USA

at Illmo~s Institute of Technology,

Chrcago.

13 July 1979)

ned uvlth the effect of recycle on reactor stabdity [ I-31 Typically, m these studies Peclet number IS held constant as recycte ratio IS vaned Thrs leads to compansons of stabmty charactensttcs of reactors of different residence tune drstnbutlon (RTD) vanances In effect. this van&on IS the same as varymg the Peclet number for a TRAM wnhout recycle In both cases the vmance (6 of the RTD changes with the vanation m either PecIet number or

1450

Shorter

Commumcatlons

recycle ratlo from some nununum value (zero for Pe = m and R = 0) to one for zero Peclet number or mfimte recycle ratio In thu paper, the tubular reactor vessel with axml mlxmg and recycle (TRAM-R) IS analyzed to establish Its RTD ExpressIons for the moments of the RTD are developed as function of recycle ratio and Peclet number It IS shown that RTD vmance can be held constant while varying either recycle ratio or Peclet number arbttranly The second parameter value IS then determined from the vananLe equation With thu two-parameter model, effects of recycle on a TRAM can be mvestlgated without altermg the vanance of the recycle system RTD No stabrhty analysts IS mcluded m this work

Ct Fig The transform evaluated passage time dlstnbutlon

ANALYSIS

1 Recycle

system

at z = 1 IS the transform of the li-st (RTD without recycle loop)

The recycle system with the TRAM IS shown m F@ 1 If we put a mass, M,. of tracer mstantaneously into the inlet stream of the vessel itself, we can wnte the material balance on tracer in the TRAM m drmenslonless form as (9) $-Pe(l+R)&-Pe(l+R)E=O In order to find the RTD of the TRAM-R we must consider the response to a tracer mlectlon of M external to the recycle loop[4] A matenal balance at the mixing pomt on tracer then gves In dlmenslonless form

where KL u(l+ R)t c( =-,c=D’ L

r=I/L,Pe=The boundary

condmons

for a closed

vessel

Gmc, Ml v

S(t) f Rq = (1 + R)c,

are

where c, and c, are shown eqn(9) yields

at z=O,-$+Pe(l+R)c=Pe(l+R)g(=) where

6( Q ) 1s the Duac delta function

m Fm

1 Convoluting

(10) eqn (10) wrth

S(t)*cn+Rc,*c~=(l+R)c,*cR

and at and since

dc=o

z=]

The nunal comhtron

’a2

IS

then c(0, z) = 0

cn f Rc, * CR= (I + R)c,

(4)

In this formulation. the Peclet number IS formulated on a velocity which IS the volumetnc flow rate of fresh fluid only divided by the vessel cross-sectional area Takmg the Laplace transform of eqn (1) wdh respect to a , we obtam

Takmg the Laplace

of eqn (11) and solvmg

become WI= t/(4P(ti(l+ (6)

&=o The solution

to the equation

set (So-()

f=

for cf [4]

Now E(S) 1s the transform of the RTD of the system However, En and 5,~ are not exactly the same The dtierence IS m the tune scales smce for Es, Q IS the tune vanable of integration m the transformation If we subsutute (sL/u(l+R)) for s mto the definmon of m, then

$$-Pe(l+R)%-Pe(l+R)sE=O after using the in&al coinhtion The transformed boundary condmons

transform

(11)

IS

[_p(y!)

-

R))

s)

where f= L/u IS the mean residence tune of the system. Then we can obtam & from eqn (9) wnh the new definmon of m Followmg van der Laan[5], the mean and vmance of the system RTD may be found duectly from the Laplace transform (eqn 12). Hrlth the mod&d eqn (9) subshtuted

exp(-mm)exp[(~)z]+P(~)exp[(~)z] (8) [(yy-(9)2exp(-*)I

where P=Pe(l+R) and m = t/(Pe’ (1 + R)’ + 4Pe ( I+ R)s)

Shorter Communmauons The resultmg of evaluatmg

eqn (13) and eqn (14) IS fi, = r

&+c&=--

(15)

(~~R)[(1+*~)+2/P-~+~exp(-P)

I (16)

The result for ~1, (eqn 15) LSm agreement wth the closed ends condmon at the boundanes In demenslonless form &R+I IiR

I+R

L-L+zexp(-P) [ P P= P=

I

(17)

where agam P = (I + R) Pe and Pe IS the Peclet number based on the fresh flmd velocny, (IL/D) P can be considered the Peclet number based on the actual reactor velocity, (1 + R)u For detzuls of thu development see Mathur[61 We may Inspect the behavior of &L m eqn (17) as the vanables tend to lmnts As P approaches rnfhnty-plug Row, the $ value approaches R/(1 + R), the vanance of a plug flow wdh recycle system As P approaches zero or as R appraoches infinity, d approaches 1. the mlxed reactor vanance As R approaches zero, d=$-j$+j$exQ(-p)

ship (eqn 17). analysis can be carned out m which some property of the reactor system, I e exit concentratron or reactor stahhty. IS related to P or R for a gwen RTD vanance In a gross way, this can be used to determine the relative effects of large scale nuxmg and smaller scale ddTuslonal type mlxmg on the reactor property under mvestrgahon The Laplace transform of the RTD was Inverted for 3 values of uz Several sets of P-R values consistent w-nh that value of u* were chosen for the mverslon This was done m order to demonstrate how the shape of the RTD curve vaned wuh changmg recycle at constant d The results for the 3 values of u’, 0 25.0 5 and 0 75 are shown m Figs 2-4, respecbvely The behavior m all three figures IS consistent and It illustrates an mterestmg pomt The peak value IS smallest for R = 0 (Pe at the mmunum possible value) and mcreases and the peak itself becomes more symmetnc as R mcreases As R becomes large a second peak begms to appear at a value of time approximately double that of the first peak t This changmg RTD demonstrates the shift m behavior from that of a tram to that of a plug flow with recycle as R varies from zero to infimty

t

(18)

the value for TRAM mthout recycle as mven by van der Laan In order to obtam the RTD m the tune domam, eqn (12) must be inverted A closed form mverslon could not be obtamed Inversion was however accomplished for 3 speclfrc cases by usmg the method of numerical quadrature given by Bellman et of [7] In this method an Integral IS replaced by a finite sum, I e

I

0’f(r) dr - 8

WJtr,)

145 1

;\

R=O281

p=400

EC8

(19)

where W, and r, are adjustable parameters To make eqn (19) exact, they described a connectlon between numerical quadrature and orthogonal polynomlak and showed that

’A-d dx = g, W~cc)

I0

(20)

where the x, are now the zeroes of the shifted Legendre polynomials Pn*(x) and the W, are a set of wet&s We can wnte B(s) =

f-0

em’@E(B) d(B) where B = r/r= tu/L

I

(21)

8 Fig 2 Residence

and usmg a new vanable of mtegratlon, quadrature formula of eqn (20) to obtam z

WI&- &i)-

bme dlstnbutlon,

2

3

TRAM-R, 2 = 0 25

x = e@, we apply the

B(s)

where g(x) = E( -Inx) Knowmg J?(s) at different values s, E(@,) IS obtzuned The detalkd development IS gnfen m[6]

(22) of

RESULTSAND MSCUSSION The pnmary result of this paper IS eqn (17) which relates the vanance of the system RTD to the system Peclet number, P, and the recycle ratio, R If a tubular reactor 1s described by Its RTD vanance alone as IS often the case, then there 1s an mfimte set of P and R values which yreld that value of 2 With this relation-

tThe curves are dashed to m&cate mterpolauon between points near the second peak because m the numerical mverslon method the values of time at whmh the curve ordmate was evaluated could not be chosen arb&anly The addmona! peaks must actually be present smce the calculated points cannot be smoothly connected wnhout them

E(9)

0

I

FU 3 Residence

2

3 8 tune dmtnbutlon. TRAM-R, ~3 = 0 50

Shorter Commumcatlons NUI’ATION

dlmenslonless tracer concentratron dlspersron coefficient residence time dlstnbution distance along vessel reactor vessel length parameter defined below eqn (8) peclet number based on fresh feed flow rate peclet number based on total vessel flow rate recycle ratio laplace transform vmable fluid velocity time time constant, L./U dimensionless distance along vessel, I/L dtmenslordess time t/r(l + R) dtrac delta function first moment of E _m*& dimensional variance of ti c? dimensionless variance of E

c

1

2

I

0

3

REBERENCES

8

Fig 4 Residence

ttme drstnbutlons

Acknowledgement-This ENG 74-13520

by NSF Grant No

VIJAY K MATHUR

Ctty Uruversrty of New York New York NY 10031. USA

Chlcal En#ma~ Pcrgunon Press Ud

TRAM-R, u* = 0 75

work was supported

Electronrc Associates Inc West Long Branch NJ 07764, USA

[l] Redly M J and Schnutz R A, A ICh E 3 1%6 22 153 [Z] McGowm C R and Perlmutter D D , A I Ch EJ 1971 17 831 [31 Perlmutter D D, Stab&y of Chemrcal Reactors PrenticeHall, Englewood Cliffs, New Jersey 1972 [41 Fu B , Wemstem H , Bcmstem B and ShafFer A B , Zna! Engng Chem Proc Des Lkv 197110 501 [51 van der Laan E T , Chem Engng SCI 1958 7 187 WI Mathur V K , Micromuung effects on reactor dynamic response Ph D Thesis, Ilho~s Institute of Technology, Clucago 1977 [71 Bellman R E, Kalaba R G and Locket J, Numerical Znuersron of fhe Laplace Transform Amencan Elsevrer, New York 1966

HERBERT

WEINSTEIN

Scwnce Vol 35 pi 1452 1454 1980 Pnntcd m Great tIntam

Note on drop

formation at low velocity in quiescent liquids (Received

10 Aprd 1979)

Drop formatlon has continuously been the subject of research III relation to mdustnal processes Numerous design problems associated wth, for example, hqmd-hqmd extraction. d&dlatlon, Aotatlon. spray combustion. spray evaporation, heat transfer eqmpment and emulslfymg systems are intimately connected wth the physics govemtng drop formation Specifically, the accurate prediction of mterfaclal area resulting from drop formation IS requved for calculatmg heat and mass transfer effects wth confidence in processes of the type mentloned above The subject of drop formation has been extensively reviewed and discussed m Refs [l-4] From these works, it may be concluded that there IS a dlshnct lack of mfonnatron concerning the fluid mechamcs of drop formatlon at low flow rates m liquni-liquid systems Although the problem has been alleviated m part by more recent studies (among others, Refs [7] and [8]), accurate final sue predlcfions of drops detachmg from nozzles are dtfficult to obtain Practlcalty all studies attemptmg to predict detached drop dtameters are based on the coupling of observed experimental results with theoretical considerations involving the forces present durmg drop formation and detachment The theoretical approaches vary in degree of detad and complexity wth no agreement emcrgmg among researchers regarding the

manner m which inertial effects in the force balance should he included The ObJectwe of this note IS to provide a simple but useful correlation for predicting detached drop diameters in hquidliquid systems at low flow rates and m which the host fluid IS quiescent Even though the comIation should only apply to drop formation at low Reynolds number based on nozzle velocity (Re G 100). it IS found that the addition of a Reynolds number dependent term allows its extension to higher flow rates (Re=MOO), while yielding droplet diameters to within an accuracy better than 7% on average DERIVATloN OF THE CORREL.ATlON Expenmental evdence From expenmental results at IS known that a typIcal plot of leadmg edge drop velocity vs ttme IS as mdlcated m Fig 1 The figure shows three stages which, from left to right, correspond to (I) Spherical growth, dtstortlon effects are small (II) Nechng. dIstortIon dlects arc large (III) Detachment and fall At low flow rates, mert~al and drag effects on the drop are small and the force balance is prunardy between Interfacial and